Have you got your health checked? We have to go the hospital to get your medical certificate.

after the 5 the period, let's go out. I'll write down in the official trip book. I have to enter the information in Neice, which is a program for the employee in our field.
After that let's go to the immigration office to get the alien registration card. Then we should go to the cell phone agency to open your cell phone. Then we need to go to a mart to buy a 21 inch TV. We have to buy a 21 inch TV for about 140,000 won.
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Are the expressions grammatical? Would you check them?

Have you got your health checked? We have to go the hospital to get your medical certificate. sounds fine

after the 5 the period, let's go out.
I don't understand this at all.

I'll write down in the official trip book. I have to enter the information in Neice, which is a program for the employee in our field.
Write what down in the official trip book?

After that let's go to the immigration office to get the alien registration card. sounds fine

Then we should go to the cell phone agency to open your cell phone.
... to open your cell phone account.

Then we need to go to a mart to buy a 21 inch TV. We have to buy a 21 inch TV for about 140,000 won.
It's better to say "we need to go to a store..." and when you're referring to the size of the TV, you need to hyphenate the measurement: "a 21-inch TV"